A336027 The least k such that i*k + 1 is a product of i different primes for i = 1..n.

1, 10, 268, 7576, 652726, 913180816

Offset: 1

Author

Vladimir Letsko, Jul 05 2020

Keywords

• nonn
• more

Comments

Since for k = 5000000000420503488, i*k+1 is a product of i different primes for i = 1..7, a(7) <= 5000000000420503488.

a(n) is the least parameter k such that equation tau(x^k) = x has at least A005425(n) solutions in positive integers.

Examples

			268 is in the sequence because 268 + 1 is prime, 2*268 + 1 is a product of 2 different primes, 3*268 + 1 is a product of 3 different primes, and 268 is the least number with such properties.
		

Links

• Vladimir Letsko, Problem K7 (in Russian).

Crossrefs

Cf. A000005, A005425, A336028.

Programs

Maple

nn := 1; for kk to 6 do
n := nn; do n := nextprime(n); f := true;
for k from 2 to kk do a := k*(n-1)+1; if not IsSquareFree(a) or nops(ifactors(a)[2]) <> k then f := false; break end if end do;
if f then nn := n-1; print(nn); break end if end do end do

PARI

isok(k,n) = {for (i=1, n, if ((omega(i*k+1) != i) || (bigomega(i*k+1) != i), return (0));); return (1);}
a(n) = {my(k=1); while(!isok(k,n), k++); k;} \\ Michel Marcus, Jul 15 2020